RESEARCH NOTE
NONLINEAR SENSITIVITY ANALYSIS OF RCMRF CONSIDERING BOTH AXIAL AND
FLEXURE EFFECTS
A.R. Habibi
Department of Civil Engineering, University of Kurdistan P.O. Box 416, Sanandaj, Iran
(Received: February 07, 2010 – Accepted in Revised Form: September 15, 2011)
doi:10.5829/idosi.ije.2011.24.03a.02
Abstract Sensitivity analysis of structures considering their nonlinear behavior has recently been an important process in optimal design of them based on new design concepts such as Performance- Based Design (PBD). The main objective of this research is to develop an efficient and applicable method for nonlinear sensitivity analysis of Reinforcement Concrete Moment Resisting Frames (RCMRF) considering both axial and flexure effects under seismic loading. For this purpose, sensitivity equations are firstly derived based on pushover method, which is a powerful tool for nonlinear analysis of structures in the PBD context, and then a procedure for nonlinear sensitivity analysis is proposed. The results of the method are compared with nonlinear sensitivity results considering only flexural effect and those of Finite Difference Method (FDM). It is shown that existence of axial forces can affect sensitivity coefficients with respect to the design variables.
Keywords Nonlinear, Sensitivity, Analysis, Pushover, RCMRF, PBD, Axial and Flexure Effects, Seismi
1. INTRODUCTION
Considerable part of computational effort in an optimization problem is usually allocated to the sensitivity analysis. Each sensitivity coefficient defines the amount of change in a structural response due to a unit change in a design variable, such as sensitivity of displacement to change in cross-sectional dimensions or reinforcement ratio.
In recent years, new design concepts such as performance-based design have become a necessity for design of new structures. Since
nonlinear methods such as pushover analysis are used in these new design concepts, it is necessary that efficient nonlinear sensitivity methods be employed in optimum design.
While considerable research effort has been put on developing the nonlinear sensitivity analysis techniques, there are a few research works in the literature that have focused on the theory of sensitivity analysis for nonlinear-framed structures.
Ryu et al. proposed a general nonlinear sensitivity analysis considering geometric and material nonlinearity [1]. They presented the formulation
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K & . 2&0 &! '&< J & & % ! HI 9 ) 20 ! E% F % G)0 ! , -.
( & % & & % $ & & *% *21 %D % # "/ C9D % ;9 * 'B!
@A &7!& &<- * & 1 &%) % 2 '! % " .)0 ! <- 9 * 2 !*% *@% :% / 9 :2! ;9 # "7 2*% ? * %=8 6 % % > !5%
4)& . %" % 1"+2 2!% % "/ )0 ! , -. ( % % $ ) )+ * ) % ( %
@9 &L ;9 K & 9 & ' &<- @% &:% K ;9 % % 5
for a truss example. Choi and Santos developed variational formulations for nonlinear design sensitivity analysis [2]. Gopalakrishna and Greimann differentiated the equilibrium equation in each Newton-Raphson iteration to obtain incremental gradients and this method was used for the nonlinear sensitivity analysis of the plane trusses [3]. Santos and Choi presented a unified approach for shape sensitivity analysis of trusses and beams considering both geometric and material nonlinearities [4]. Ohsaki and Arora presented an accumulative and incremental algorithm for the design sensitivity analysis of elastoplastic structures including geometrical nonlinearity [5]. They performed the sensitivity analysis of trusses but they reported that the method is extremely time consuming for large structures. Lee and Arora developed design sensitivity analysis of structural systems with elastoplastic material behavior using the continuum formulation and performed sensitivity analysis of a truss and a plate by this technique [6].
Barthold and Stein presented a continuum mechanical-based formulation for the variational sensitivity analysis considering nonlinear hyper elastic material behavior using either the Lagrangian or Eulerian description [7]. Szewczyk
and Ahmed presented a hybrid
numerical/neurocomputing strategy for evaluation of sensitivity coefficients of composite panels subjected to combined thermal and mechanical loads [8]. Yamazaki suggested a direct sensitivity analysis technique for finding incremental sensitivities of the path-dependent nonlinear problem based on the updated Lagrangian formulation [9]. Employing this method, they performed the sensitivity analysis of a plate.
Bugeda et al. proposed a direct formulation for computing the structural shape sensitivity analysis with a nonlinear constitutive material model [10].
It was reported that their proposed approach was valid for some specific nonlinear material models.
Schwarz and Ramm proposed the variational direct method for sensitivity analysis of structural response considering geometrical and material nonlinearity with Prantel-Reuss plasticity model [11]. Gong et al. presented a procedure for sensitivity analysis of planar steel moment frameworks with geometric and material nonlinearity [12]. In their work, analytical
formulations defining the sensitivity of displacement were derived. Habibi A.R. and Moharrami H. developed a comprehensive formulation for sensitivity analysis of planar RCMRFs considering material nonlinearity and P- Δ effects under pushover analysis based on Newton-Raphson method [13]. In their research, only flexure effect was considered to derive sensitivity equations.
The objective of present research is to develop the nonlinear theory of sensitivity analysis for RC frames considering both axial and flexure effects.
Also, sensitivity equations are modified to conclude effect of seismic loading that depends on design variables. To do this, pushover analysis that is a simplified nonlinear analysis is employed. The sensitivity equations are derived based on the analysis with incremental-load form. A three-story RCMRF has been used as an example to illustrate the applicability and efficiency of the developed sensitivity formulations.
2. NONLINEAR ANALYSIS
The first step of design optimization is the sensitivity analysis that in turn depends on the method of structural analysis. The simplest recommended method for nonlinear analysis is pushover method which is used in this study. This method of analysis that is recommended by FEMA273 [14] and ATC40 [15] is a popular tool for evaluation of seismic performance of existing and new structures. A proper material behavior model for elements and seismic loads for pushover analysis are explained in the next three sections.
2.1. Moment Curvature Relation The moment- curvature relation of every Reinforced Concrete (RC) structural element has a definitive effect on the behavior of the structure. In this research the tri-linear moment curvature relation, as shown in Figure 1, is used for expressing the nonlinear behavior of reinforced concrete sections.
To determine moment-curvature relation of RC members, some assumptions should be made that best fits with the test results. In this study, considering the effect of axial force in moment- curvature relation, the following limitations for
various states of behavior of RC elements are used [16,17]:
(a) Cracking state
h y Nd 6
I f
Mcr r (1)
h y
E fr c
cr
(2)
(b) Yielding state
1 β ηn 2 η p η 2βαp
d b 0.5f
My c t 2 0
(3)
1 k
d / cεφy y (4)
(c) Ultimate state
0
yu 1.24 0.15p 0.5n M
M (5)
R]
S) ε [(R
1.05β
φu 1 u 2 0.5 (6)
where,
);
)d/(1.7f ρf
ε E ρ (
R u s y c S ρεuEsβ1dd/(0.85fc); fr is the modulus of rupture of concrete; fc is the cylinder strength of concrete; fy is the yield strength of steel;Ec is the modulus of elasticity of concrete; Es is the modulus of elasticity of steel;
u is the ultimate strain of concrete; y is the yield strain of steel; 1 is a coefficient that depends on the strength of concrete; N is the axial force; d is the depth to rebar;
d is the cover depth for compression bars; h is the overall height of section; c is the cover to steel centroid; bt is the top width of section; y is the distance from the
neutral axis of the section to the extreme fiber in tension and is obtained from (7); Iis the moment of inertia of the section and is obtained from Equation 8. Other parameters are defined from Equations 9 to 16.
bt b h t (n 1)(A A )
d A d A 1) (n ) t (h 0.5b t y 0.5b
s s sc b
t
s s sc 2 2 b 2 t
(7)
s 2
2 s sc
2
b 3 b 2 t
3 t
) d (y A y) (d A 1) (n y) 0.5t (0.5h
t h b /12 t) (h b 0.5t) t(y b /12 t b I
(8)
(9)
0.7
0 y y 0
y ε
ε φ d /ε ε 1
η 0.75
(10)
β 1ε ε φ d β α 1
y y
y
(11)
d /b ρ A d;
/b
ρAs b s b (12)
c y c
y/f ;p ρf /f ρf
p (13)
/d
βd (14)
bd N/f
n0 c (15)
0.3 0.05 n p 0.84 1.05 0.45
c 0
(16)
where nsc is the ratio of the modulus of elasticity of steel to that of concrete; 0is the strain at maximum strength of concrete; bb is the width of the section at the bottom; t is the flange thickness of T or L beam; As is the area of bottom bars and As is the area of top bars. For the case of positive moment, Equations 1 to 16 are valid for all types of sections. For rectangular sections these equations can be simplified by substituting
b b
bb t and t0. Also, for the case of negative
Figure 1. Tri-linear moment curvature curve.
2 2
sc
2 sc
2 2
t t
b b
sc t
b
4α
1 1 1 t
α 2α d
t t
d
b b
d b b
b
t t
d b d
′ ′ ′
p+p + + p − p+p k≤
′ ′
′
− + − −
n (ρ ρ) 1 k>
k= n (ρ+ρ)+ −1 +2n (ρ+ρ β)+ −1
( ) (p β ) ( )
moment, these equations can be used by substituting bt, bb, ht, As and As instead of bb, bt, t, As and As, respectively. Ignoring axial effect in beam elements, N, which has been appeared in some above equations, can be considered to be equal to zero. Now by having the moment-curvature relations, the flexural stiffness can be specified for ends of the element as follow:
pm crp crp
p M /φ EI
EI (17)
pm crp
yp crp yp
p (M M )/(φ φ ) EI
EI (18)
pm yp
up yp up
p (M M )/(φ φ ) EI
EI (19)
where Equations 17-19 are used for zones of 1, 2 and 3 of M curve, respectively. In these equations,EIpand EIpm are the stiffness and minimum stiffness of any section; Mcrp, Myp and Mup are the cracking, yielding and ultimate moments; and crp, yp and up are corresponding curvatures.
2.2. Material Nonlinearity In this paper, a spread plasticity model that has been proposed by park et al. is utilized [18]. This plasticity model, that considers the cracking behavior of RC, can also consider material nonlinearity of RC elements with a very good approximation. It has been implemented in IDARK software [18]. In this model, for an element subjected to earthquake loading, the distribution of curvature follows Figures 2a,b which is considered for the flexibility distribution in the RC elements, where EIA and EIB are the flexural stiffness of the section at end
‘A’ and ‘B’, respectively, and EI0 is the initial stiffness of the element; A and B are the yield penetration coefficients and L is the length of the element.
Adopting the above plasticity model for RC elements, the tangent stiffness matrix of each element can be derived. General derivation of tangent stiffness matrix is given by Park et al. [18].
Considering a RC element with six degrees of freedom, with its rigid parts, as shown in Figure 3, the tangent stiffness matrix of element can be
found as follows:
b a
te K K
K (20)
where Ka and Kb are axial stiffness matrix and bending stiffness matrix, respectively and must be assembled in the element stiffness matrixKte. The element stiffness matrix calculated from Equation 20 needs to be transformed to global system:
e te T e
e T K T
K (21)
where Te is the transformation matrix of the element.
Ka can be determined from:
0
1 1
1 1
a
a AK
L
K EA
(22)
where EA/L is the axial stiffness of the element, A is the area of cross section and Ka0 is a constant matrix. Kb, is obtained from the following equation:
T E S E
b R K R
K (23)
(a)
(b)
Figure 2. (a) Curvature distribution along a RC element and (b) Flexibility assumption along a RC element.
where:
~1
; ~ 1 / 1 0 / 1
0 / 1 1 /
1
K LK L
L L
L
RTE L S S (24)
where L is length of element and L~
can be obtained from the following equation:
B B
A 1 A
A B
A B B
A λ 1 λ
λ λ L 1
λ ; λ 1
λ λ 1 λ λ 1
L~ 1 ~
(25) where A and B are the portions of rigid zone at the element ends. The elements of KS in (24) are obtained from the following equation:
b a S11 S22 b c S11 S21 S12
e b B A 0 S11
/f f K K
; /f f K K K
D f L EI EI 12EI
K (26)
where:
L L
L f f f D
S S
S f
S S S f
S S
S f
B A c
b a e
B B A
A c
B B B A
b
B A A A a
) 1
(
;
2 2
2
4 6 4
4 6 4
2 2
3 2 3 3 2 2 1
3 2 3
3 2 1
3 3 3 2 2
1
(27)
where
B A
1 EI EI
S ; S2
EI0EIA
EIB; and
0 B
A3 EI EI EI
S .
The yield penetration parameters used in Equation 27, specify the portion of the element where it cracks. When the moment of the section under consideration is less than cracking moment, the values of these parameters are considered to be equal to zero. For the single curvature of the element, when its moments are greater than cracking moment, the values of the parameters are considered to be equal to 0.5. For other cases, assuming linear moment distribution along the element, these parameters can be determined from:
p crp A B pm
p max min M M M M ,1 ,α
α (28)
where subscript p specifies any general point p and cr represents the cracking state under consideration. pm is the maximum yield penetration parameter, obtained in previous load steps. Usually, the flexural stiffness at the center of the element is equal to elastic stiffness and is determined from Equation 29. For the case of single curvature if moments are greater than cracking moments, their values can be modified by Equation 30.
0 0
0 0 0
2
B A
B A
EI EI
EI EI EI
(29)
B A
B A
EI EI
EI EI EI
2
0 (30)
where EIA0 and EIB0 are the elastic stiffness at ends ‘A’ and ‘B’, respectively; and EIA and EIB Figure 3. Rigid zone and ends definitions.
are the inelastic stiffness at ends ‘A’ and ‘B’, respectively. For a member that yield penetration spreads over the whole element
whenAB 1
, EI0 is obtained from Equation 30. In this case, A and B and initial flexibility are modified to capture the actual flexibility distribution. These modifications are:B B A A
B B B A
A f f f f
f f f
f
0 0
0
(31)
A
B
1 (32)
A A A
A f f
f
f0 0 / (33)
where:
0 0 1/
; / 1
; /
1 EI f EI f EI
fA A B B (34)
2.3. Seismic Loading In the pushover analysis, the load vector must incrementally be increased. At each load step, the base shear increment is applied to the structure with a predefined profile over the height of the structure. The incremental lateral load vector can be computed as:
Cv Vb ns cv
cv cv Vb
PE .
, 2 ,
1 ,
.
(35)
where Vb is the incremental base shear and Cv is the vector of lateral load distribution factors
s numberof stories
cv,s 1,..., , which is determined using the following relation [14]:
ns ns s
s
k Hs Ws
k Hs Ws s
cv , 1,...,
1
,
(36)
where Ws is the portion of the building seismic weight at story level s; Hs is the vertical distance from base of the building to story level s; ns is the number of stories; and k is a parameter that
has been recommended by FEMA273 as follows [14]:
5 . 2 2
5 . 2 5 . 0 75
. 0 5 . 0
5 . 0 1
T T T
T
k (37)
where T is the fundamental period of the building.
2.4. Nonlinear Responses Since in nonlinear analysis of structure, stiffness isn’t constant and depends on the internal forces of elements and internal forces depend on displacement field, it will be a function of displacement field; on the other hand, displacement depends on stiffness. Then structural analysis is involved in incremental process. The equilibrium equation of the structure is solved using the simple incremental method considering several load steps. No iteration is required within one load step, and thus the numerical algorithm is generally well behaved and exhibits good computational efficiency. In each load step following equation must be solved [19]:
l l l
T u P
K (38)
where KTl is the tangent stiffness of the structure at load step l; ul is the incremental displacement vector of the structure at load step land Pl is the external load vector imposed on the structure at load step l. By having the incremental displacement vector from Equation 38, the displacement vector of the structure at each load step must be updated as follows:
l l
l u u
u 1 (39)
where ul1 and ul are the displacement vector of the structure at load steps (l1) and (l), respectively. Considering equilibrium of forces for an element, the incremental internal forces of the element at load step l can be determined from Equation 40:
l e l e l
e K u
F
(40)
where Kel is the tangent stiffness matrix of the
element and ule is the incremental displacement vector of the element at load step l. By having the incremental internal forces from last mentioned equation, the internal forces of the element at load step l must be updated as follows:
l e l e l
e F F
F 1 (41)
where
e1l
F and
el
F are the internal force vectors of the element at load steps (l1) and (l), respectively.
3. NONLINEAR SENSITIVITY ANALYSIS In the structures undergoing inelastic behavior, incremental methods are usually used to solve equilibrium equation. In this case, displacements are nonlinear function of external loads. Also stiffness is a function of displacements and forces. This causes sensitivity coefficients such as displacement sensitivities to be dependent on stiffness sensitivity, displacement sensitivity, external load sensitivity and internal load sensitivity; while in linear structures, they only depend on stiffness sensitivity and external load sensitivity. Finding sensitivity relations in nonlinear structures is too difficult for this complicacy; on the other hand, applying linear sensitivity theory leads to incorrect results. In the present study, to drive relations for nonlinear sensitivities, equilibrium equations are differentiated at each load step. This is explained in the next section.
3.1. Displacement Sensitivity Considering incremental Equation 38 and differentiating it with respect to design variablexj, we get to:
j l j
l l T l j l T
dx P d dx
u K d dx u
dK (42)
where KTl is the global tangent stiffness matrix;
ul
is the incremental displacement vector;
j l T
dx dK
is the sensitivity of the global tangent stiffness
matrix;
j l
dx du
is the sensitivity of the incremental displacement vector and
j l
dx dP
is the sensitivity of the incremental external load vector at load step l with respect to design variablexj. In this study, Equation 42 is used as the basic formulation for obtaining the incremental displacement sensitivities. Assuming that all terms in Equation 42 except
j l
dx du
are known, it will be rearranged to find:
l
j l T j l l
T j
l
dx u dK dx
P K d
dx u
d 1
(43) The total sensitivity of displacement vector can be obtained by summing up the sensitivities in all load steps as follows:
j l j
l j l
dx u d dx du dx
du 1 (44)
where
j l
dx du1
and
j l
dx
du are the sensitivities of the incremental displacement vector at load steps
) 1
(l and (l), respectively. The term
j l
dx du
is the sensitivity of the incremental displacement vector which is obtained from Equation 43. For this, sensitivities of stiffness and external loads need to be determined. In the coming sections, the way each sensitivity item in Equation 43 should be obtained, is explained.
3.1.1. Sensitivity of stiffness The tangent stiffness matrix of the structure at each load step is obtained by properly assembling of the tangent stiffness matrix of elements:
ne
e e l
T K
K
1
(45) where ne is the number of structural elements and
Ke is the tangent stiffness matrix of the element with material nonlinearity. Differentiating Equation
45 with respect to any generalized variable, xj, sensitivity of the tangent stiffness matrix can be obtained from:
ne
e j
e j
l T
dx dK dx
dK
1 ) (
(46) Based on the last equation, to compute tangent stiffness sensitivity, it is sufficient to calculate sensitivity of tangent stiffness of all elements.
Sensitivity of tangent stiffness matrix of each element is obtained by differentiating from Equation 21:
e j T te e j
e T
dx T dK dx
dK (47)
In the above equation,
j te
dx
dK is the sensitivity of the tangent stiffness matrix of the element including axial effects sensitivity and bending effects sensitivity in local system and can be obtained by differentiating from Equation 20:
j b j a j te
dx dK dx dK dx
dK (48)
It is noted that the first and second terms in the right hand side of Equation 48 must be properly assembled. The first term represents the axial stiffness while the second one represents flexural stiffness. These two sensitivity matrix are obtained by differentiating Equations 22 and 23 noting that matrices of a 0
K
and E R
utilized in these equations are independent of design variables and can be determined as follow:
0 a j j
a K
dx dA dx
dK (49)
T E j E S j
b R
dx R dK dx
dK (50)
Assuming rigid diaphragm for floors, there will be no change in axial forces of beams. Accordingly, the axial stiffness sensitivity for beams is zero. For column elements by definingAbh, the sensitivity of A with respect to b ish, and with respect to h isb. It is zero with respect to other variables. The
sensitivity of matrix KS utilized in Equation 50 is determined by differentiating from Equation 24:
j S j
S S j j S
dx L K d L dx L
K Ld L dx K
L d dx
dK 1 1 ~1
~
~
~
~ ~
(51)
where:
j B j B
j A j A
j
j A j B
j A j
B
j B j A B A j
dx d dx d
dx d dx d
dx L d
dx d dx d
dx d dx d dx L
d dx d dx
L d
~1
~ 1
~ 1
(52)
In Equation 52, sensitivity of A with respect to hA is1/2L; also sensitivity of B with respect to hB is1/2L. For a typical beam hA and hB have been shown in Figure 3. Similar definitions can be made for columns between two floor beams. In Equation 51 sensitivities of matrix KS with respect to any design variable can be obtained by differentiation of its elements as follow:
e b j e b j
j b e
B A A
j B
B j B A A j e b j S
D f L dx f dD dx
L d
dx L df D
EI EI EI EI
dx EI dEI
EI dx EI EI dEI dx EI
dEI D
L f dx dK
0 0
0 0 11
12 12
(53)
2 11 11
11 12
b S c j b b S j c b c j S j S
f K f dx df f K dx df f f dx dK dx
dK (54)
2 11 11
11 22
b S a j b b S j a b a j S j
S
f K f dx df f K dx df f f dx dK dx
dK (55)
Where
1 3 3 2
2
2 3 3 2 2
4 3
8 6
3 4
6
S dx S
S d
dx S d dx S
df
B j
A A A
j B B A
A A j
a
(56)
1 3 2 2
3
2 2 3 2 3
4 3
8 6
3 4
6
S dx S
S d
dx S d dx S
df
A j
B B B
j A A B
B B j
b
(57)
1 2
3 3 2 3
2 2
3 2 2
2 3
4 2
3 4 2
dx S S d
S
dx S d
dx S df
j B B B B
B
j A A A A
A j
c
(58)
j c b a
j c c a j b b j a j e
dx L L d f f f
dx L f df dx f
f df dx df dx dD
2
2
2
2
(59)
dx L d dx d dx
L d
j b j a
j
(60) where:
j A B A
j B j
j B A B
j A j
A j
B B j
A
dx EI dEI EI dx EI
dEI dx S dEI
dx EI dEI EI dx EI
dEI dx
S dEI
dx EI EI dEI dx S dEI
0 0
3
0 0
2 1
(61)
In these equations, the derivative of EI0 with respect to any variable X can be obtained by differentiating from Equation 29 with respect to X.
This leads to Equation 62:
j B j
A B
A B A
A
j
B
B
j
